Non-abelian anyons and some cousins of the Arad-Herzog conjecture

Long ago, Arad and Herzog (AH) conjectured that, in finite simple groups, the product of two conjugacy classes of length greater than one is never a single conjugacy class. We discuss implications of this conjecture for non-abelian anyons in 2 + 1-dimensional discrete gauge theories. Thinking in this way also suggests closely related statements about finite simple groups and their associated discrete gauge theories. We prove these statements and provide some physical intuition for their validity. Finally, we explain that the lack of certain dualities in theories with non-abelian finite simple gauge groups provides a non-trivial check of the AH conjecture.

PAL: A Framework for Physically Assisted Learning Through Design and Exploration With a Haptic Robot Buddy

Robots are an opportunity for interactive and engaging learning activities. In this paper we consider the premise that haptic force feedback delivered through a held robot can enrich learning of science-related concepts by building physical intuition as learners design experiments and physically explore them to solve problems they have posed. Further, we conjecture that combining this rich feedback with pen-and-paper interactions, e.g., to sketch experiments they want to try, could lead to fluid interactions and benefit focus. However, a number of technical barriers interfere with testing this approach, and making it accessible to learners and their teachers. In this paper, we propose a framework for Physically Assisted Learning based on stages of experiential learning which can guide designers in developing and evaluating effective technology, and which directs focus on how haptic feedback could assist with design and explore learning stages. To this end, we demonstrated a possible technical pathway to support the full experience of designing an experiment by drawing a physical system on paper, then interacting with it physically after the system recognizes the sketch, interprets as a model and renders it haptically. Our proposed framework is rooted in theoretical needs and current advances for experiential learning, pen-paper interaction and haptic technology. We further explain how to instantiate the PAL framework using available technologies and discuss a path forward to a larger vision of physically assisted learning.

Computer-Assisted Methods for Analyzing Periodic Orbits in Vibrating Gravitational Billiards

Using rigorous numerical methods, we prove the existence of 608 isolated periodic orbits in a gravitational billiard in a vibrating unbounded parabolic domain. We then perform pseudo-arclength continuation in the amplitude of the parabolic surface’s oscillation to compute large, global branches of periodic orbits. These branches are themselves proven rigorously using computer-assisted methods. Our numerical investigations strongly suggest the existence of multiple pitchfork bifurcations in the billiard model. Based on the numerics, physical intuition and existing results for a simplified model, we conjecture that for any pair [Formula: see text], there is a constant [Formula: see text] for which periodic orbits consisting of [Formula: see text] impacts per period [Formula: see text] cannot be sustained for amplitudes of oscillation below [Formula: see text]. We compute a verified upper bound for the conjectured critical amplitude for [Formula: see text] using our rigorous pseudo-arclength continuation.

The Relationship between Entropy and the Reliability Function in Application of Physical and Statistical Approach to Nanosystems

The paper considers application of the physical and statistical approach to the issue of nanosystems reliability. A general method of solving the main equation in this approach is suggested and the solution in quadratures is obtained in one-dimensional stationary case. It is used to study the behaviour of entropy and the reliability function under certain assumptions. The cases of constant, linear, and quadratic degradation rates are analysed. In the first two cases the results correspond to physical intuition while in the last case (quadratic rate) the formal solution demonstrates counterintuitive behaviour. Numerical correlations between the distribution entropy dynamics and the reliability function are given.

The Quantum Cookbook

The Quantum Cookbook provides a unique bridge between popular exposition and formal textbook presentation, based on derivations of the iconic equations of quantum mechanics. This is a book for curious readers with some background in physics and sufficient mathematical capability. It aims not to teach readers how to do quantum mechanics but rather helps them to understand how to think about quantum mechanics. Each derivation is presented as a ‘recipe’ with listed ingredients, including standard results from the mathematician’s toolkit, set out in a series of easy-to-follow steps. The recipes have been written sympathetically, for readers who—like the author—will often struggle to follow the logic of a derivation which misses out steps that are ‘obvious’, or which use techniques that readers are assumed to know. The simple truth is that quantum mechanics did not suddenly materialize overnight in the minds of its creators, fully formed, complete with all its axioms and principles. It was instead tortured from much more familiar classical physical descriptions, over a period of decades, as physicists struggled to interpret a series of ever more baffling experimental results. Only later was a much higher level of abstraction introduced into quantum mechanics, in an attempt to establish a secure mathematical foundation that would eradicate all the confusing classical misconceptions inherited from its birth and early childhood. Whilst there are some obvious exceptions, for the most part these derivations are triumphs of physical intuition over mathematical rigour and consistency.

Less may be more: an informed reflection on molecular descriptors for drug design and discovery

The phenomenal advances of machine learning in the context of drug design have led to the development of a plethora of molecular descriptors. And yet, there might be value in using just a handful of them – inspired by our physical intuition.